Uniform integrability conjecture for irreducible lattices generated by norm-three vectors

Let $\Lambda$ be an irreducible integral lattice generated by vectors of norm $3$. A lattice is $\sigma$-integrable if its bilinear form, after multiplication by $\sigma$, admits an integral standard-lattice realization. Uniform integrability conjecture. There exists a positive integer $\sigma$ such that for any irreducible integral lattice $\Lambda$ generated by vectors of norm $3$, the lattice $\Lambda$ is $\sigma$-integrable. This is a lattice-theoretic uniformity question related to the integrability of lattices arising from graphs with smallest eigenvalue at least $-3$; no resolution is supplied in the source context.